Ch 6,Risk and Rates
of Return
Return
Risk
? 2002,Prentice Hall,Inc,
Chapter 6,Objectives
? Inflation and rates of return
? How to measure risk
(variance,standard deviation,beta)
? How to reduce risk
(diversification)
? How to price risk
(security market line,CAPM)
Inflation,Rates of Return,
and the Fisher Effect
Interest
Rates
Interest Rates Conceptually,
Interest Rates Conceptually,
Nominal
risk-free
Interest
Rate
krf
Interest Rates Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Interest Rates Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
Interest Rates Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
+
Interest Rates Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
+
Inflation-
risk
premium
IRP
Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
+
Inflation-
risk
premium
IRP
Mathematically,
Interest Rates
Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
+
Inflation-
risk
premium
IRP
Mathematically,
(1 + krf) = (1 + k*) (1 + IRP)
Interest Rates
Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
+
Inflation-
risk
premium
IRP
Mathematically,
(1 + krf) = (1 + k*) (1 + IRP)
This is known as the,Fisher Effect”
Interest Rates
? Suppose the real rate is 3%,and the nominal
rate is 8%,What is the inflation rate
premium?
(1 + krf) = (1 + k*) (1 + IRP)
(1.08) = (1.03) (1 + IRP)
(1 + IRP) = (1.0485),so
IRP = 4.85%
Interest Rates
Term Structure of Interest Rates
? The pattern of rates of return for debt
securities that differ only in the length
of time to maturity,
Term Structure of Interest Rates
? The pattern of rates of return for debt
securities that differ only in the length
of time to maturity,
yield
to
maturity
time to maturity (years)
Term Structure of Interest Rates
? The pattern of rates of return for debt
securities that differ only in the length
of time to maturity,
yield
to
maturity
time to maturity (years)
Term Structure of Interest Rates
yield
to
maturity
time to maturity (years)
? The yield curve may be downward
sloping or,inverted” if rates are
expected to fall,
Term Structure of Interest Rates
yield
to
maturity
time to maturity (years)
? The yield curve may be downward
sloping or,inverted” if rates are
expected to fall,
For a Treasury security,what is
the required rate of return?
For a Treasury security,what is
the required rate of return?
Required
rate of
return
=
For a Treasury security,what is
the required rate of return?
Since Treasuries are essentially free of
default risk,the rate of return on a
Treasury security is considered the
“risk-free” rate of return,
Required
rate of
return
=
Risk-free
rate of
return
For a corporate stock or bond,what is
the required rate of return?
For a corporate stock or bond,what is
the required rate of return?
Required
rate of
return
=
For a corporate stock or bond,what is
the required rate of return?
Required
rate of
return
=
Risk-free
rate of
return
For a corporate stock or bond,what is
the required rate of return?
How large of a risk premium should we
require to buy a corporate security?
Required
rate of
return
= +
Risk-free
rate of
return
Risk
premium
Returns
? Expected Return - the return that an
investor expects to earn on an asset,
given its price,growth potential,etc,
? Required Return - the return that an
investor requires on an asset given
its risk and market interest rates,
Expected Return
State of Probability Return
Economy (P) Orl,Utility Orl,Tech
Recession,20 4% -10%
Normal,50 10% 14%
Boom,30 14% 30%
For each firm,the expected return on the
stock is just a weighted average,
Expected Return
State of Probability Return
Economy (P) Orl,Utility Orl,Tech
Recession,20 4% -10%
Normal,50 10% 14%
Boom,30 14% 30%
For each firm,the expected return on the
stock is just a weighted average,
k = P(k1)*k1 + P(k2)*k2 +,..+ P(kn)*kn
Expected Return
State of Probability Return
Economy (P) Orl,Utility Orl,Tech
Recession,20 4% -10%
Normal,50 10% 14%
Boom,30 14% 30%
k = P(k1)*k1 + P(k2)*k2 +,..+ P(kn)*kn
k (OU) =,2 (4%) +,5 (10%) +,3 (14%) = 10%
Expected Return
State of Probability Return
Economy (P) Orl,Utility Orl,Tech
Recession,20 4% -10%
Normal,50 10% 14%
Boom,30 14% 30%
k = P(k1)*k1 + P(k2)*k2 +,..+ P(kn)*kn
k (OI) =,2 (-10%)+,5 (14%) +,3 (30%) = 14%
Based only on your
expected return
calculations,which
stock would you
prefer?
RISK?
Have you considered
What is Risk?
? The possibility that an actual return
will differ from our expected return,
? Uncertainty in the distribution of
possible outcomes,
What is Risk?
? Uncertainty in the distribution of
possible outcomes,
What is Risk?
? Uncertainty in the distribution of
possible outcomes,
0
0, 0 5
0, 1
0, 1 5
0, 2
0, 2 5
0, 3
0, 3 5
0, 4
0, 4 5
0, 5
4 8 12
Company A
return
What is Risk?
? Uncertainty in the distribution of
possible outcomes,
return
0
0, 0 2
0, 0 4
0, 0 6
0, 0 8
0, 1
0, 1 2
0, 1 4
0, 1 6
0, 1 8
0, 2
- 1 0 -5 0 5 10 15 20 25 30
Company B
0
0, 0 5
0, 1
0, 1 5
0, 2
0, 2 5
0, 3
0, 3 5
0, 4
0, 4 5
0, 5
4 8 12
Company A
return
How do we Measure Risk?
? To get a general idea of a stock’s
price variability,we could look at
the stock’s price range over the
past year,
52 weeks Yld Vol Net
Hi Lo Sym Div % PE 100s Hi Lo Close Chg
134 80 IBM,52,5 21 143402 98 95 9549 -3
115 40 MSFT … 29 558918 55 52 51 94 -475
How do we Measure Risk?
? A more scientific approach is to
examine the stock’s standard
deviation of returns,
? Standard deviation is a measure of
the dispersion of possible outcomes,
? The greater the standard deviation,
the greater the uncertainty,and
therefore,the greater the risk,
Standard Deviation
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
( 4% - 10%)2 (.2) = 7.2
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
( 4% - 10%)2 (.2) = 7.2
(10% - 10%)2 (.5) = 0
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
( 4% - 10%)2 (.2) = 7.2
(10% - 10%)2 (.5) = 0
(14% - 10%)2 (.3) = 4.8
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
( 4% - 10%)2 (.2) = 7.2
(10% - 10%)2 (.5) = 0
(14% - 10%)2 (.3) = 4.8
Variance = 12
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
( 4% - 10%)2 (.2) = 7.2
(10% - 10%)2 (.5) = 0
(14% - 10%)2 (.3) = 4.8
Variance = 12
Stand,dev,= 12 =
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
( 4% - 10%)2 (.2) = 7.2
(10% - 10%)2 (.5) = 0
(14% - 10%)2 (.3) = 4.8
Variance = 12
Stand,dev,= 12 = 3.46%
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
(-10% - 14%)2 (.2) = 115.2
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
(-10% - 14%)2 (.2) = 115.2
(14% - 14%)2 (.5) = 0
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
(-10% - 14%)2 (.2) = 115.2
(14% - 14%)2 (.5) = 0
(30% - 14%)2 (.3) = 76.8
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
(-10% - 14%)2 (.2) = 115.2
(14% - 14%)2 (.5) = 0
(30% - 14%)2 (.3) = 76.8
Variance = 192
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
(-10% - 14%)2 (.2) = 115.2
(14% - 14%)2 (.5) = 0
(30% - 14%)2 (.3) = 76.8
Variance = 192
Stand,dev,= 192 =
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
(-10% - 14%)2 (.2) = 115.2
(14% - 14%)2 (.5) = 0
(30% - 14%)2 (.3) = 76.8
Variance = 192
Stand,dev,= 192 = 13.86%
= (ki - k)2 P(ki) s
n
i=1
S
Which stock would you prefer?
How would you decide?
Which stock would you prefer?
How would you decide?
Orlando Orlando
Utility Technology
Expected Return 10% 14%
Standard Deviation 3.46% 13.86%
Summary
It depends on your tolerance for risk!
Remember,there’s a tradeoff between
risk and return,
It depends on your tolerance for risk!
Remember,there’s a tradeoff between
risk and return,
Return
Risk
It depends on your tolerance for risk!
Remember,there’s a tradeoff between
risk and return,
Return
Risk
Portfolios
? Combining several securities
in a portfolio can actually
reduce overall risk,
? How does this work?
Suppose we have stock A and stock B,
The returns on these stocks do not tend
to move together over time (they are
not perfectly correlated),
rate
of
return
time
Suppose we have stock A and stock B,
The returns on these stocks do not tend
to move together over time (they are
not perfectly correlated),
rate
of
return
time
kA
Suppose we have stock A and stock B,
The returns on these stocks do not tend
to move together over time (they are
not perfectly correlated),
rate
of
return
time
kA
kB
What has happened to the
variability of returns for the
portfolio?
rate
of
return
time
kA
kB
rate
of
return
time
kp
kA
kB
What has happened to the
variability of returns for the
portfolio?
Diversification
? Investing in more than one security
to reduce risk,
? If two stocks are perfectly positively
correlated,diversification has no
effect on risk,
? If two stocks are perfectly negatively
correlated,the portfolio is perfectly
diversified,
? If you owned a share of every stock
traded on the NYSE and NASDAQ,
would you be diversified?
YES!
? Would you have eliminated all of
your risk?
NO! Common stock portfolios still
have risk,
Some risk can be diversified
away and some cannot,
? Market risk (systematic risk) is
nondiversifiable,This type of risk
cannot be diversified away,
? Company-unique risk (unsystematic
risk) is diversifiable,This type of risk
can be reduced through
diversification,
Market Risk
? Unexpected changes in interest rates,
? Unexpected changes in cash flows
due to tax rate changes,foreign
competition,and the overall business
cycle,
Company-unique Risk
? A company’s labor force goes on
strike,
? A company’s top management dies
in a plane crash,
? A huge oil tank bursts and floods a
company’s production area,
As you add stocks to your portfolio,
company-unique risk is reduced,
As you add stocks to your portfolio,
company-unique risk is reduced,
portfolio
risk
number of stocks
As you add stocks to your portfolio,
company-unique risk is reduced,
portfolio
risk
number of stocks
Market risk
As you add stocks to your portfolio,
company-unique risk is reduced,
portfolio
risk
number of stocks
Market risk
company-
unique
risk
Do some firms have more
market risk than others?
Yes,For example,
Interest rate changes affect all firms,
but which would be more affected,
a) Retail food chain
b) Commercial bank
Yes,For example,
Interest rate changes affect all firms,
but which would be more affected,
a) Retail food chain
b) Commercial bank
Do some firms have more
market risk than others?
? Note
As we know,the market compensates
investors for accepting risk - but
only for market risk,Company-
unique risk can and should be
diversified away,
So - we need to be able to measure
market risk,
This is why we have Beta,
Beta,a measure of market risk,
? Specifically,beta is a measure of how
an individual stock’s returns vary
with market returns,
? It’s a measure of the,sensitivity” of
an individual stock’s returns to
changes in the market,
? A firm that has a beta = 1 has average
market risk,The stock is no more or less
volatile than the market,
? A firm with a beta > 1 is more volatile than
the market,
The market’s beta is 1
? A firm that has a beta = 1 has average
market risk,The stock is no more or less
volatile than the market,
? A firm with a beta > 1 is more volatile than
the market,
– (ex,technology firms)
The market’s beta is 1
? A firm that has a beta = 1 has average
market risk,The stock is no more or less
volatile than the market,
? A firm with a beta > 1 is more volatile than
the market,
– (ex,technology firms)
? A firm with a beta < 1 is less volatile than
the market,
The market’s beta is 1
? A firm that has a beta = 1 has average
market risk,The stock is no more or less
volatile than the market,
? A firm with a beta > 1 is more volatile than
the market,
– (ex,technology firms)
? A firm with a beta < 1 is less volatile than
the market,
– (ex,utilities)
The market’s beta is 1
Calculating Beta
Calculating Beta
-5 -15 5 10 15
-15
-10
-10
-5
5
10
15
XYZ Co,returns
S&P 500
returns
Calculating Beta
-5 -15 5 10 15
-15
-10
-10
-5
5
10
15
XYZ Co,returns
S&P 500
returns
.,,,
.,,,
.,,,,,,,
.,,,
.,,,
.,,,
.,,,
.,,
.,,,
.,,,
Calculating Beta
-5 -15 5 10 15
-15
-10
-10
-5
5
10
15
XYZ Co,returns
S&P 500
returns
.,,,
.,,,
.,,,,,,,
.,,,
.,,,
.,,,
.,,,
.,,
.,,,
.,,,
Calculating Beta
-5 -15 5 10 15
-15
-10
-10
-5
5
10
15
XYZ Co,returns
S&P 500
returns
.,,,
.,,,
.,,,,,,,
.,,,
.,,,
.,,,
.,,,
.,,
.,,,
.,,,
Beta = slope
= 1.20
Summary,
? We know how to measure risk,using
standard deviation for overall risk
and beta for market risk,
? We know how to reduce overall risk
to only market risk through
diversification,
? We need to know how to price risk so
we will know how much extra return
we should require for accepting extra
risk,
What is the Required Rate of
Return?
? The return on an investment
required by an investor given
market interest rates and the
investment’s risk,
Required
rate of
return
=
Required
rate of
return
= +
Risk-free
rate of
return
Required
rate of
return
= +
Risk-free
rate of
return
Risk
premium
market
risk
Required
rate of
return
= +
Risk-free
rate of
return
Risk
premium
market
risk company- unique risk
Required
rate of
return
= +
Risk-free
rate of
return
Risk
premium
market
risk company- unique risk
can be diversified
away
Required
rate of
return
= +
Risk-free
rate of
return
Risk
premium
Required
rate of
return
Beta
Let’s try to graph this
relationship!
Required
rate of
return
,
Risk-free
rate of
return
(6%)
Beta
12%
1
Required
rate of
return
,
Risk-free
rate of
return
(6%)
Beta
12%
1
security
market
line
(SML)
This linear relationship between
risk and required return is
known as the Capital Asset
Pricing Model (CAPM),
Required
rate of
return
,
Risk-free
rate of
return
(6%)
Beta
12%
1
SML
0
Required
rate of
return
,
Risk-free
rate of
return
(6%)
Beta
12%
1
SML
0
Is there a riskless
(zero beta) security?
Required
rate of
return
Beta
,12%
1
SML
0
Is there a riskless
(zero beta) security?
Treasury
securities are
as close to riskless
as possible,Risk-free rate of
return
(6%)
Required
rate of
return
,
Beta
12%
1
SML Where does the S&P 500
fall on the SML?
Risk-free
rate of
return
(6%)
0
Required
rate of
return
,
Beta
12%
1
SML Where does the S&P 500
fall on the SML?
The S&P 500 is
a good
approximation
for the market
Risk-free
rate of
return
(6%)
0
Required
rate of
return
,
Beta
12%
1
SML
Utility
Stocks
Risk-free
rate of
return
(6%)
0
Required
rate of
return
,
Beta
12%
1
SML High-tech
stocks
Risk-free
rate of
return
(6%)
0
The CAPM equation,
kj = krf + j (km - krf )
b
The CAPM equation,
kj = krf + j (km - krf )
where,
kj = the required return on security
j,
krf = the risk-free rate of interest,
j = the beta of security j,and
km = the return on the market index,
The CAPM equation,
b
b
Example,
? Suppose the Treasury bond rate is
6%,the average return on the
S&P 500 index is 12%,and Walt
Disney has a beta of 1.2,
? According to the CAPM,what
should be the required rate of
return on Disney stock?
kj = krf + (km - krf )
kj =,06 + 1.2 (.12 -,06)
kj =,132 = 13.2%
According to the CAPM,Disney
stock should be priced to give a
13.2% return,
b
Required
rate of
return
,
Beta
12%
1
SML
0
Risk-free
rate of
return
(6%)
Required
rate of
return
,
Beta
12%
1
SML
0
Theoretically,every
security should lie
on the SML
Risk-free
rate of
return
(6%)
Required
rate of
return
,
Beta
12%
1
SML
0
Theoretically,every
security should lie
on the SML
If every stock
is on the SML,
investors are being fully
compensated for risk,Risk-free rate of
return
(6%)
Required
rate of
return
,
Beta
12%
1
SML
0
If a security is above
the SML,it is
underpriced,
Risk-free
rate of
return
(6%)
Required
rate of
return
,
Beta
12%
1
SML
0
If a security is above
the SML,it is
underpriced,
If a security is
below the SML,it
is overpriced,Risk-free
rate of
return
(6%)
Simple Return Calculations
Simple Return Calculations
t t+1
$50 $60
Simple Return Calculations
= = 20% Pt+1 - Pt 60 - 50 P
t 50
t t+1
$50 $60
Simple Return Calculations
= = 20% Pt+1 - Pt 60 - 50 P
t 50
- 1 = -1 = 20% Pt+1 60 P
t 50
t t+1
$50 $60
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00
Feb $63.80
Mar $59.00
Apr $62.00
May $64.50
Jun $69.00
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80
Mar $59.00
Apr $62.00
May $64.50
Jun $69.00
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00
Apr $62.00
May $64.50
Jun $69.00
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00
May $64.50
Jun $69.00
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50
Jun $69.00
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00 0.000
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00 0.000
Aug $75.00 0.087
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00 0.000
Aug $75.00 0.087
Sep $82.50 0.100
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00 0.000
Aug $75.00 0.087
Sep $82.50 0.100
Oct $73.00 -0.115
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00 0.000
Aug $75.00 0.087
Sep $82.50 0.100
Oct $73.00 -0.115
Nov $80.00 0.096
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00 0.000
Aug $75.00 0.087
Sep $82.50 0.100
Oct $73.00 -0.115
Nov $80.00 0.096
Dec $86.00 0.075
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160 0.049
Feb $63.80 0.100 0.049
Mar $59.00 -0.075 0.049
Apr $62.00 0.051 0.049
May $64.50 0.040 0.049
Jun $69.00 0.070 0.049
Jul $69.00 0.000 0.049
Aug $75.00 0.087 0.049
Sep $82.50 0.100 0.049
Oct $73.00 -0.115 0.049
Nov $80.00 0.096 0.049
Dec $86.00 0.075 0.049
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160 0.049 0.012321
Feb $63.80 0.100 0.049 0.002601
Mar $59.00 -0.075 0.049 0.015376
Apr $62.00 0.051 0.049 0.000004
May $64.50 0.040 0.049 0.000081
Jun $69.00 0.070 0.049 0.000441
Jul $69.00 0.000 0.049 0.002401
Aug $75.00 0.087 0.049 0.001444
Sep $82.50 0.100 0.049 0.002601
Oct $73.00 -0.115 0.049 0.028960
Nov $80.00 0.096 0.049 0.002090
Dec $86.00 0.075 0.049 0.000676
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160 0.049 0.012321
Feb $63.80 0.100 0.049 0.002601
Mar $59.00 -0.075 0.049 0.015376
Apr $62.00 0.051 0.049 0.000004
May $64.50 0.040 0.049 0.000081
Jun $69.00 0.070 0.049 0.000441
Jul $69.00 0.000 0.049 0.002401
Aug $75.00 0.087 0.049 0.001444
Sep $82.50 0.100 0.049 0.002601
Oct $73.00 -0.115 0.049 0.028960
Nov $80.00 0.096 0.049 0.002090
Dec $86.00 0.075 0.049 0.000676
0.0781 St,Dev,sum,divide by (n-1),and take sq root,
Calculator solution using HP 10B,
? Enter monthly return on 10B calculator,
followed by sigma key (top right corner),
? Shift 7 gives you the expected return,
? Shift 8 gives you the standard deviation,
of Return
Return
Risk
? 2002,Prentice Hall,Inc,
Chapter 6,Objectives
? Inflation and rates of return
? How to measure risk
(variance,standard deviation,beta)
? How to reduce risk
(diversification)
? How to price risk
(security market line,CAPM)
Inflation,Rates of Return,
and the Fisher Effect
Interest
Rates
Interest Rates Conceptually,
Interest Rates Conceptually,
Nominal
risk-free
Interest
Rate
krf
Interest Rates Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Interest Rates Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
Interest Rates Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
+
Interest Rates Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
+
Inflation-
risk
premium
IRP
Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
+
Inflation-
risk
premium
IRP
Mathematically,
Interest Rates
Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
+
Inflation-
risk
premium
IRP
Mathematically,
(1 + krf) = (1 + k*) (1 + IRP)
Interest Rates
Conceptually,
Nominal
risk-free
Interest
Rate
krf
=
Real
risk-free
Interest
Rate
k*
+
Inflation-
risk
premium
IRP
Mathematically,
(1 + krf) = (1 + k*) (1 + IRP)
This is known as the,Fisher Effect”
Interest Rates
? Suppose the real rate is 3%,and the nominal
rate is 8%,What is the inflation rate
premium?
(1 + krf) = (1 + k*) (1 + IRP)
(1.08) = (1.03) (1 + IRP)
(1 + IRP) = (1.0485),so
IRP = 4.85%
Interest Rates
Term Structure of Interest Rates
? The pattern of rates of return for debt
securities that differ only in the length
of time to maturity,
Term Structure of Interest Rates
? The pattern of rates of return for debt
securities that differ only in the length
of time to maturity,
yield
to
maturity
time to maturity (years)
Term Structure of Interest Rates
? The pattern of rates of return for debt
securities that differ only in the length
of time to maturity,
yield
to
maturity
time to maturity (years)
Term Structure of Interest Rates
yield
to
maturity
time to maturity (years)
? The yield curve may be downward
sloping or,inverted” if rates are
expected to fall,
Term Structure of Interest Rates
yield
to
maturity
time to maturity (years)
? The yield curve may be downward
sloping or,inverted” if rates are
expected to fall,
For a Treasury security,what is
the required rate of return?
For a Treasury security,what is
the required rate of return?
Required
rate of
return
=
For a Treasury security,what is
the required rate of return?
Since Treasuries are essentially free of
default risk,the rate of return on a
Treasury security is considered the
“risk-free” rate of return,
Required
rate of
return
=
Risk-free
rate of
return
For a corporate stock or bond,what is
the required rate of return?
For a corporate stock or bond,what is
the required rate of return?
Required
rate of
return
=
For a corporate stock or bond,what is
the required rate of return?
Required
rate of
return
=
Risk-free
rate of
return
For a corporate stock or bond,what is
the required rate of return?
How large of a risk premium should we
require to buy a corporate security?
Required
rate of
return
= +
Risk-free
rate of
return
Risk
premium
Returns
? Expected Return - the return that an
investor expects to earn on an asset,
given its price,growth potential,etc,
? Required Return - the return that an
investor requires on an asset given
its risk and market interest rates,
Expected Return
State of Probability Return
Economy (P) Orl,Utility Orl,Tech
Recession,20 4% -10%
Normal,50 10% 14%
Boom,30 14% 30%
For each firm,the expected return on the
stock is just a weighted average,
Expected Return
State of Probability Return
Economy (P) Orl,Utility Orl,Tech
Recession,20 4% -10%
Normal,50 10% 14%
Boom,30 14% 30%
For each firm,the expected return on the
stock is just a weighted average,
k = P(k1)*k1 + P(k2)*k2 +,..+ P(kn)*kn
Expected Return
State of Probability Return
Economy (P) Orl,Utility Orl,Tech
Recession,20 4% -10%
Normal,50 10% 14%
Boom,30 14% 30%
k = P(k1)*k1 + P(k2)*k2 +,..+ P(kn)*kn
k (OU) =,2 (4%) +,5 (10%) +,3 (14%) = 10%
Expected Return
State of Probability Return
Economy (P) Orl,Utility Orl,Tech
Recession,20 4% -10%
Normal,50 10% 14%
Boom,30 14% 30%
k = P(k1)*k1 + P(k2)*k2 +,..+ P(kn)*kn
k (OI) =,2 (-10%)+,5 (14%) +,3 (30%) = 14%
Based only on your
expected return
calculations,which
stock would you
prefer?
RISK?
Have you considered
What is Risk?
? The possibility that an actual return
will differ from our expected return,
? Uncertainty in the distribution of
possible outcomes,
What is Risk?
? Uncertainty in the distribution of
possible outcomes,
What is Risk?
? Uncertainty in the distribution of
possible outcomes,
0
0, 0 5
0, 1
0, 1 5
0, 2
0, 2 5
0, 3
0, 3 5
0, 4
0, 4 5
0, 5
4 8 12
Company A
return
What is Risk?
? Uncertainty in the distribution of
possible outcomes,
return
0
0, 0 2
0, 0 4
0, 0 6
0, 0 8
0, 1
0, 1 2
0, 1 4
0, 1 6
0, 1 8
0, 2
- 1 0 -5 0 5 10 15 20 25 30
Company B
0
0, 0 5
0, 1
0, 1 5
0, 2
0, 2 5
0, 3
0, 3 5
0, 4
0, 4 5
0, 5
4 8 12
Company A
return
How do we Measure Risk?
? To get a general idea of a stock’s
price variability,we could look at
the stock’s price range over the
past year,
52 weeks Yld Vol Net
Hi Lo Sym Div % PE 100s Hi Lo Close Chg
134 80 IBM,52,5 21 143402 98 95 9549 -3
115 40 MSFT … 29 558918 55 52 51 94 -475
How do we Measure Risk?
? A more scientific approach is to
examine the stock’s standard
deviation of returns,
? Standard deviation is a measure of
the dispersion of possible outcomes,
? The greater the standard deviation,
the greater the uncertainty,and
therefore,the greater the risk,
Standard Deviation
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
( 4% - 10%)2 (.2) = 7.2
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
( 4% - 10%)2 (.2) = 7.2
(10% - 10%)2 (.5) = 0
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
( 4% - 10%)2 (.2) = 7.2
(10% - 10%)2 (.5) = 0
(14% - 10%)2 (.3) = 4.8
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
( 4% - 10%)2 (.2) = 7.2
(10% - 10%)2 (.5) = 0
(14% - 10%)2 (.3) = 4.8
Variance = 12
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
( 4% - 10%)2 (.2) = 7.2
(10% - 10%)2 (.5) = 0
(14% - 10%)2 (.3) = 4.8
Variance = 12
Stand,dev,= 12 =
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Utility,Inc,
( 4% - 10%)2 (.2) = 7.2
(10% - 10%)2 (.5) = 0
(14% - 10%)2 (.3) = 4.8
Variance = 12
Stand,dev,= 12 = 3.46%
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
(-10% - 14%)2 (.2) = 115.2
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
(-10% - 14%)2 (.2) = 115.2
(14% - 14%)2 (.5) = 0
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
(-10% - 14%)2 (.2) = 115.2
(14% - 14%)2 (.5) = 0
(30% - 14%)2 (.3) = 76.8
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
(-10% - 14%)2 (.2) = 115.2
(14% - 14%)2 (.5) = 0
(30% - 14%)2 (.3) = 76.8
Variance = 192
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
(-10% - 14%)2 (.2) = 115.2
(14% - 14%)2 (.5) = 0
(30% - 14%)2 (.3) = 76.8
Variance = 192
Stand,dev,= 192 =
= (ki - k)2 P(ki) s
n
i=1
S
Orlando Technology,Inc,
(-10% - 14%)2 (.2) = 115.2
(14% - 14%)2 (.5) = 0
(30% - 14%)2 (.3) = 76.8
Variance = 192
Stand,dev,= 192 = 13.86%
= (ki - k)2 P(ki) s
n
i=1
S
Which stock would you prefer?
How would you decide?
Which stock would you prefer?
How would you decide?
Orlando Orlando
Utility Technology
Expected Return 10% 14%
Standard Deviation 3.46% 13.86%
Summary
It depends on your tolerance for risk!
Remember,there’s a tradeoff between
risk and return,
It depends on your tolerance for risk!
Remember,there’s a tradeoff between
risk and return,
Return
Risk
It depends on your tolerance for risk!
Remember,there’s a tradeoff between
risk and return,
Return
Risk
Portfolios
? Combining several securities
in a portfolio can actually
reduce overall risk,
? How does this work?
Suppose we have stock A and stock B,
The returns on these stocks do not tend
to move together over time (they are
not perfectly correlated),
rate
of
return
time
Suppose we have stock A and stock B,
The returns on these stocks do not tend
to move together over time (they are
not perfectly correlated),
rate
of
return
time
kA
Suppose we have stock A and stock B,
The returns on these stocks do not tend
to move together over time (they are
not perfectly correlated),
rate
of
return
time
kA
kB
What has happened to the
variability of returns for the
portfolio?
rate
of
return
time
kA
kB
rate
of
return
time
kp
kA
kB
What has happened to the
variability of returns for the
portfolio?
Diversification
? Investing in more than one security
to reduce risk,
? If two stocks are perfectly positively
correlated,diversification has no
effect on risk,
? If two stocks are perfectly negatively
correlated,the portfolio is perfectly
diversified,
? If you owned a share of every stock
traded on the NYSE and NASDAQ,
would you be diversified?
YES!
? Would you have eliminated all of
your risk?
NO! Common stock portfolios still
have risk,
Some risk can be diversified
away and some cannot,
? Market risk (systematic risk) is
nondiversifiable,This type of risk
cannot be diversified away,
? Company-unique risk (unsystematic
risk) is diversifiable,This type of risk
can be reduced through
diversification,
Market Risk
? Unexpected changes in interest rates,
? Unexpected changes in cash flows
due to tax rate changes,foreign
competition,and the overall business
cycle,
Company-unique Risk
? A company’s labor force goes on
strike,
? A company’s top management dies
in a plane crash,
? A huge oil tank bursts and floods a
company’s production area,
As you add stocks to your portfolio,
company-unique risk is reduced,
As you add stocks to your portfolio,
company-unique risk is reduced,
portfolio
risk
number of stocks
As you add stocks to your portfolio,
company-unique risk is reduced,
portfolio
risk
number of stocks
Market risk
As you add stocks to your portfolio,
company-unique risk is reduced,
portfolio
risk
number of stocks
Market risk
company-
unique
risk
Do some firms have more
market risk than others?
Yes,For example,
Interest rate changes affect all firms,
but which would be more affected,
a) Retail food chain
b) Commercial bank
Yes,For example,
Interest rate changes affect all firms,
but which would be more affected,
a) Retail food chain
b) Commercial bank
Do some firms have more
market risk than others?
? Note
As we know,the market compensates
investors for accepting risk - but
only for market risk,Company-
unique risk can and should be
diversified away,
So - we need to be able to measure
market risk,
This is why we have Beta,
Beta,a measure of market risk,
? Specifically,beta is a measure of how
an individual stock’s returns vary
with market returns,
? It’s a measure of the,sensitivity” of
an individual stock’s returns to
changes in the market,
? A firm that has a beta = 1 has average
market risk,The stock is no more or less
volatile than the market,
? A firm with a beta > 1 is more volatile than
the market,
The market’s beta is 1
? A firm that has a beta = 1 has average
market risk,The stock is no more or less
volatile than the market,
? A firm with a beta > 1 is more volatile than
the market,
– (ex,technology firms)
The market’s beta is 1
? A firm that has a beta = 1 has average
market risk,The stock is no more or less
volatile than the market,
? A firm with a beta > 1 is more volatile than
the market,
– (ex,technology firms)
? A firm with a beta < 1 is less volatile than
the market,
The market’s beta is 1
? A firm that has a beta = 1 has average
market risk,The stock is no more or less
volatile than the market,
? A firm with a beta > 1 is more volatile than
the market,
– (ex,technology firms)
? A firm with a beta < 1 is less volatile than
the market,
– (ex,utilities)
The market’s beta is 1
Calculating Beta
Calculating Beta
-5 -15 5 10 15
-15
-10
-10
-5
5
10
15
XYZ Co,returns
S&P 500
returns
Calculating Beta
-5 -15 5 10 15
-15
-10
-10
-5
5
10
15
XYZ Co,returns
S&P 500
returns
.,,,
.,,,
.,,,,,,,
.,,,
.,,,
.,,,
.,,,
.,,
.,,,
.,,,
Calculating Beta
-5 -15 5 10 15
-15
-10
-10
-5
5
10
15
XYZ Co,returns
S&P 500
returns
.,,,
.,,,
.,,,,,,,
.,,,
.,,,
.,,,
.,,,
.,,
.,,,
.,,,
Calculating Beta
-5 -15 5 10 15
-15
-10
-10
-5
5
10
15
XYZ Co,returns
S&P 500
returns
.,,,
.,,,
.,,,,,,,
.,,,
.,,,
.,,,
.,,,
.,,
.,,,
.,,,
Beta = slope
= 1.20
Summary,
? We know how to measure risk,using
standard deviation for overall risk
and beta for market risk,
? We know how to reduce overall risk
to only market risk through
diversification,
? We need to know how to price risk so
we will know how much extra return
we should require for accepting extra
risk,
What is the Required Rate of
Return?
? The return on an investment
required by an investor given
market interest rates and the
investment’s risk,
Required
rate of
return
=
Required
rate of
return
= +
Risk-free
rate of
return
Required
rate of
return
= +
Risk-free
rate of
return
Risk
premium
market
risk
Required
rate of
return
= +
Risk-free
rate of
return
Risk
premium
market
risk company- unique risk
Required
rate of
return
= +
Risk-free
rate of
return
Risk
premium
market
risk company- unique risk
can be diversified
away
Required
rate of
return
= +
Risk-free
rate of
return
Risk
premium
Required
rate of
return
Beta
Let’s try to graph this
relationship!
Required
rate of
return
,
Risk-free
rate of
return
(6%)
Beta
12%
1
Required
rate of
return
,
Risk-free
rate of
return
(6%)
Beta
12%
1
security
market
line
(SML)
This linear relationship between
risk and required return is
known as the Capital Asset
Pricing Model (CAPM),
Required
rate of
return
,
Risk-free
rate of
return
(6%)
Beta
12%
1
SML
0
Required
rate of
return
,
Risk-free
rate of
return
(6%)
Beta
12%
1
SML
0
Is there a riskless
(zero beta) security?
Required
rate of
return
Beta
,12%
1
SML
0
Is there a riskless
(zero beta) security?
Treasury
securities are
as close to riskless
as possible,Risk-free rate of
return
(6%)
Required
rate of
return
,
Beta
12%
1
SML Where does the S&P 500
fall on the SML?
Risk-free
rate of
return
(6%)
0
Required
rate of
return
,
Beta
12%
1
SML Where does the S&P 500
fall on the SML?
The S&P 500 is
a good
approximation
for the market
Risk-free
rate of
return
(6%)
0
Required
rate of
return
,
Beta
12%
1
SML
Utility
Stocks
Risk-free
rate of
return
(6%)
0
Required
rate of
return
,
Beta
12%
1
SML High-tech
stocks
Risk-free
rate of
return
(6%)
0
The CAPM equation,
kj = krf + j (km - krf )
b
The CAPM equation,
kj = krf + j (km - krf )
where,
kj = the required return on security
j,
krf = the risk-free rate of interest,
j = the beta of security j,and
km = the return on the market index,
The CAPM equation,
b
b
Example,
? Suppose the Treasury bond rate is
6%,the average return on the
S&P 500 index is 12%,and Walt
Disney has a beta of 1.2,
? According to the CAPM,what
should be the required rate of
return on Disney stock?
kj = krf + (km - krf )
kj =,06 + 1.2 (.12 -,06)
kj =,132 = 13.2%
According to the CAPM,Disney
stock should be priced to give a
13.2% return,
b
Required
rate of
return
,
Beta
12%
1
SML
0
Risk-free
rate of
return
(6%)
Required
rate of
return
,
Beta
12%
1
SML
0
Theoretically,every
security should lie
on the SML
Risk-free
rate of
return
(6%)
Required
rate of
return
,
Beta
12%
1
SML
0
Theoretically,every
security should lie
on the SML
If every stock
is on the SML,
investors are being fully
compensated for risk,Risk-free rate of
return
(6%)
Required
rate of
return
,
Beta
12%
1
SML
0
If a security is above
the SML,it is
underpriced,
Risk-free
rate of
return
(6%)
Required
rate of
return
,
Beta
12%
1
SML
0
If a security is above
the SML,it is
underpriced,
If a security is
below the SML,it
is overpriced,Risk-free
rate of
return
(6%)
Simple Return Calculations
Simple Return Calculations
t t+1
$50 $60
Simple Return Calculations
= = 20% Pt+1 - Pt 60 - 50 P
t 50
t t+1
$50 $60
Simple Return Calculations
= = 20% Pt+1 - Pt 60 - 50 P
t 50
- 1 = -1 = 20% Pt+1 60 P
t 50
t t+1
$50 $60
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00
Feb $63.80
Mar $59.00
Apr $62.00
May $64.50
Jun $69.00
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80
Mar $59.00
Apr $62.00
May $64.50
Jun $69.00
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00
Apr $62.00
May $64.50
Jun $69.00
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00
May $64.50
Jun $69.00
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50
Jun $69.00
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00 0.000
Aug $75.00
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00 0.000
Aug $75.00 0.087
Sep $82.50
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00 0.000
Aug $75.00 0.087
Sep $82.50 0.100
Oct $73.00
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00 0.000
Aug $75.00 0.087
Sep $82.50 0.100
Oct $73.00 -0.115
Nov $80.00
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00 0.000
Aug $75.00 0.087
Sep $82.50 0.100
Oct $73.00 -0.115
Nov $80.00 0.096
Dec $86.00
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160
Feb $63.80 0.100
Mar $59.00 -0.075
Apr $62.00 0.051
May $64.50 0.040
Jun $69.00 0.070
Jul $69.00 0.000
Aug $75.00 0.087
Sep $82.50 0.100
Oct $73.00 -0.115
Nov $80.00 0.096
Dec $86.00 0.075
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160 0.049
Feb $63.80 0.100 0.049
Mar $59.00 -0.075 0.049
Apr $62.00 0.051 0.049
May $64.50 0.040 0.049
Jun $69.00 0.070 0.049
Jul $69.00 0.000 0.049
Aug $75.00 0.087 0.049
Sep $82.50 0.100 0.049
Oct $73.00 -0.115 0.049
Nov $80.00 0.096 0.049
Dec $86.00 0.075 0.049
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160 0.049 0.012321
Feb $63.80 0.100 0.049 0.002601
Mar $59.00 -0.075 0.049 0.015376
Apr $62.00 0.051 0.049 0.000004
May $64.50 0.040 0.049 0.000081
Jun $69.00 0.070 0.049 0.000441
Jul $69.00 0.000 0.049 0.002401
Aug $75.00 0.087 0.049 0.001444
Sep $82.50 0.100 0.049 0.002601
Oct $73.00 -0.115 0.049 0.028960
Nov $80.00 0.096 0.049 0.002090
Dec $86.00 0.075 0.049 0.000676
(a) (b)
monthly expected
month price return return (a - b)2
Dec $50.00
Jan $58.00 0.160 0.049 0.012321
Feb $63.80 0.100 0.049 0.002601
Mar $59.00 -0.075 0.049 0.015376
Apr $62.00 0.051 0.049 0.000004
May $64.50 0.040 0.049 0.000081
Jun $69.00 0.070 0.049 0.000441
Jul $69.00 0.000 0.049 0.002401
Aug $75.00 0.087 0.049 0.001444
Sep $82.50 0.100 0.049 0.002601
Oct $73.00 -0.115 0.049 0.028960
Nov $80.00 0.096 0.049 0.002090
Dec $86.00 0.075 0.049 0.000676
0.0781 St,Dev,sum,divide by (n-1),and take sq root,
Calculator solution using HP 10B,
? Enter monthly return on 10B calculator,
followed by sigma key (top right corner),
? Shift 7 gives you the expected return,
? Shift 8 gives you the standard deviation,
